Differential entropy and tiling
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This paper relates the differential entropy of a sufficiently nice probability density functionp on Euclideann-space to the problem of tilingn-space by the translates of a given compact symmetric convex setS with nonempty interior. The relationship occurs via the concept of the epsilon entropy ofn-space under the norm induced byS, with probability induced byp. An expression is obtained for this entropy asε approaches 0, which equals the differential entropy ofp, plusn times the logarithm of 2/ε, plus the logarithm of the reciprocal of the volume ofS, plus a constantC(S) depending only onS, plus a term approaching zero withε. The constantC(S) is called the entropic packing constant ofS; the main results of the paper concern this constant. It is shown thatC(S) is between 0 and 1; furthermore,C(S) is zero if and only if translates ofS tile all ofn-space.
Key wordsDifferential entropy Tiling Entropy Close packing Random coding Convex sets Epsilon entropy Information-theoretic geometry
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